On Cramér-von Mises statistic for the spectral distribution of random matrices

Abstract

Let F N and F be the empirical and limiting spectral distributions of an N × N Wigner matrix. The Cramér-von Mises (CvM) statistic is a classical goodness-of-fit statistic that characterizes the distance between F N and F in ℓ 2 -norm. In this paper, we consider a mesoscopic approximation of the CvM statistic for Wigner matrices, and derive its limiting distribution. In the appendix, we also give the limiting distribution of the CvM statistic (without approximation) for the toy model CUE.

Statistics of extremes in eigenvalue-counting staircases

“…where c = 3 2 log log N +c . Expansion around t = 0 leads to the same coefficients C(β) k which are thus independent of (as can be seen already from (17)) and to the result for C k ( ) in (10), again related to the ones for the fBm0 on an interval given and numerically checked in [20].…”

“…max θ∈[θ A ,θ B ] |N θ A (θ) − E(N θ A (θ))|. After appropriate normalization this is simply the Kolmogorov-Smirnov (KS) statistics, an important outstanding open problem for spectra of random matrices (see recent discussion and references in [17]). First results in this direction have been obtained very recently in [18] by deriving a (weak) law of large numbers for the KS statistics in the Gaussian Unitary Ensemble (GUE).…”

“…The expectation value in (17) over the CUE β (N ) computed using (1) has the form E[ N j=1 g(θ j )] where we defined…”